How is the above discussion of regimes of validity for scalar-tensor theories relevant to the main observable we are considering in this paper, \(c_{\rm GW}\) ? This is discussed in detail in [1]} and we here summarise the essential results relevant for this paper. When deriving (REF ) and (REF ) we were implicitly working on cosmological scales, firmly within the regime of validity of (REF ). However, as one approaches the cutoff of the theory \(\lesssim \Lambda \) , the existence of a UV completion will manifest itself by additional interactions (highly suppressed on cosmological scales) becoming important. Precisely what these interactions are is determined by the (unknown) UV completion, but we can
infer some types of candidate additional interactions from investigating radiative corrections to generic Horndeski theories [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}.
The example set of such interactions focused on in [1]} is
\({\cal L}^{(n)}=M_{\rm Pl}^2 G_{\mu \nu } \frac{\Box ^n}{M^{2n+4}}(\nabla ^\mu \phi \nabla ^\nu \phi ),\) 
